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In mathematics, a set is a well-defined collection of distinct objects, called elements or members. These elements can be anything — numbers, letters, symbols, or even other sets. Sets are usually denoted by capital letters such as A,B,C,A,B,C, and their elements are written within curly brackets. For example, A={1,2,3,4}A={1,2,3,4} represents a set containing the numbers 1, 2, 3, and 4. The concept of sets forms the foundation for many areas of mathematics, including algebra, geometry, probability, and logic. Sets can be finite (having a limited number of elements) or infinite (having endlessly many elements, such as the set of all natural numbers). There are different types of sets, including empty sets, subsets, universal sets, equal sets, and disjoint sets. Operations such as union, intersection, difference, and complement allow mathematicians to combine or compare sets to form new ones. Set theory helps organize mathematical concepts and relationships clearly and logically.
Definition:
A set is a collection of well-defined and distinct objects or elements.
Notation:
Represented using curly brackets: e.g., A={2,4,6,8}A={2,4,6,8}.
Elements are separated by commas.
Types of Sets:
Finite Set: Contains a countable number of elements.
Infinite Set: Contains infinitely many elements (e.g., natural numbers).
Empty (Null) Set: Has no elements, denoted by ∅∅ or {}{}.
Subset: A set whose elements all belong to another set.
Universal Set: Contains all possible elements under discussion.
Equal Sets: Two sets with exactly the same elements.
Disjoint Sets: Sets having no common elements.
Set Operations:
Union (A∪BA∪B): Combines all elements from both sets.
Intersection (A∩BA∩B): Includes only common elements.
Difference (A−BA−B): Elements in AA that are not in BB.
Complement (A′A′): Elements not in the set AA, relative to a universal set.
Applications:
Used in probability, logic, computer science, data organization, and relations among numbers.
Helps define functions, relations, and mappings in mathematics.
1pc mathematical set
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